ON THE CHARACTERIZATION OF GENERALIZED DHOMBRES EQUATIONS HAVING NON CONSTANT LOCAL ANALYTIC OR FORMAL SOLUTIONS

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1 Annales Univ. Sci. Bdapest., Sect. Comp ON THE CHARACTERIZATION OF GENERALIZED DHOMBRES EQUATIONS HAVING NON CONSTANT LOCAL ANALYTIC OR FORMAL SOLUTIONS Ldwig Reich Graz, Astria Jörg Tomaschek Ville de Lxemborg, Grand Dchy of Lxemborg Dedicated to Professors Zoltán Daróczy and Imre Kátai on their 75th anniversary Commnicated by Antal Járai Received April 7, 203; accepted Jly 05, 203 Abstract. We discss solvability conditions of the generalized Dhombres fnctional eqation in the complex domain. Or focs lies on the so-called infinity case, bt also the z 0 case is investigated. That means that we consider soltions of a generalized Dhombres eqation with initial vale f = w 0, w 0 0, or f =, or fz 0 = for z 0 0,. For both sitations we give a characterization of the generalized Dhombres eqations which are solvable.. Introdction For a complex variable the generalized Dhombres eqation is given by. fzfz = ϕfz where the fnction ϕ is known and the fnction f is nknown. The eqation was introdced by J. Dhombres in [2] in the year 975. The complex case was first Key words and phrases: Dhombres fnctional eqation, Formal power series. 200 Mathematics Sbject Classification: 30D05, 39B32. J. Tomaschek is spported by the National Research Fnd, Lxemborg AFR , and cofnded nder the Marie Crie Actions of the Eropean Commission FP7-COFUND.

2 282 L. Reich and J. Tomaschek considered by L. Reich, J. Smítal, M. Štefánková in [3] in The athors started compting soltions f of. with f0 = 0. This was followed by frther works in the sbseqent years. Nowadays formal soltions f of. where f0 = 0, f0 = w 0 0 and fz 0 =, z 0 0 are well described. Recently the sitations with f = w 0 C \ {0} and f = were considered. In this note we are interested in the problem to find a necessary and sfficient condition in terms of the given fnction ϕ for the existence of non constant local analytic or formal soltions of. assming certain initial vales. These considerations started with a formla, namely 32 in [3] and were contined in [5] with one theorem which deals with the niversal solvability of an eqation of the form. with f0 = w 0 C\{0}, see Theorem. below. In this article we will mainly consider the remaining cases, namely fz 0 =, z 0 0 and f = w 0 C \ {0} or f =. We start with the so called infinity case. Before we do this we want to recall the existence theorem already mentioned, therefore we need some well-known transformations. In [4] the generalized Dhombres fnctional eqation. was transformed to the eqivalent linear fnctional eqation.2 w 0 + z k Uz = Uψ by fz = w 0 + gz = w 0 + T z k, g0 = 0, Uz = T z, ϕz = w ϕy w 0, ϕ0 = 0 and ψz k = ϕz k. The condition that a local analytic or formal soltion f of. is non constant is then transformed to the eqivalent condition that the linear eqation.2 has a soltion U different from 0. In [5] the following theorem was shown. Theorem.. The fnctional eqation.2 w 0 + z k Uz = Uψz with ψz zc[z k ] has a soltion U which is different from zero if and only if there exists a fnction g 0 with g 0 z = z +... sch that.3 ϕz = g 0 w0 + z k g 0 z. Hence. has a non constant soltion f with f0 = w 0 if and only if there exists g 0 z = z +... and k N sch that.3 holds. In this article we want to derive, where it is possible, analogos characterizations of the solvability of generalized Dhombres eqations as we have in Theorem.. As we will see, this is possible in the infinity case, in the case where fz 0 = and z 0 0 we get a somewhat different general condition for solvability. The difference lies in the fact that in the infinity case a generator g 0 plays the main role, while there is no generator in the z 0 case. The

3 Characterization of Dhombres eqations 283 main reason for that is that each possible soltion of. with fz 0 = for some z 0 0 is of the form fz = + gz with gz 0 = 0, g z 0 0, in a neighborhood of z = z 0. For an introdction to formal power series we refer the reader to [], by E we denote the roots of one. The ring of formal power series in one variable with the ssal addition, mltiplication and sbstittion is given by We will se the following definition C[z ] := { F F z = a ν z ν, a ν C }. ν 0 zc[z k ] := { G Gz = a νk+ z νk+}. ν 0 2. The case f = w 0 C \ {0} or f = We consider the generalized Dhombres fnctional eqation. fzfz = ϕfz with f = w 0 C \ {0}. We need to rewrite this eqation, therefore we recall some transformations recently done in [6]. Let f be holomorphic in a neighborhood of z = and let f = w 0 C \ {0}. We write z = where is in a neighborhood of zero and we get f f = ϕ f or eqivalently f = ϕ f f. We define the new fnction ˆf by ˆf = f, ˆf is holomorphic in a neighborhood of = 0. Using ˆf leads to ˆf = ϕ ˆf. ˆf

4 284 L. Reich and J. Tomaschek Since ˆf0 = w 0 it is possible to write ˆf = w 0 + g where g is holomorphic in a neighborhood of zero and g0 = 0. We get w 0 + g = ϕw 0 + g. w 0 + g For = 0 we have ϕw 0 = w 0 and hence we can write ϕ as ϕy = w 0 + ϕy w 0 with ϕ0 = 0. So we get the so-called transformed generalized Dhombres fnctional eqation 2. g w 0 + g = ϕg which is different from the sitation where f0 = w 0 C. We refer to the developed theory on generalized Dhombres eqations and therefore we write g = T k for a k N and ord T =. Hence we have or if we sbstitte T for k T w 0 + T k = ϕt k, T T k w 0 + k = ϕ k. Taking the k-th root see Lemma 2. below and comparing the first coefficients on both sides leads to T T w 0 + k = ψ where ψ k = ϕ k. Here, like in 2., in the brackets a fraction occrs, this is different to the classical theory. We set U = T and so we obtain the linear fnctional eqation 2.2 w 0 + k U = Uψ. We need the following lemma which dates back to Lemma 2 in [4], nevertheless we want to give a straighter proof. Therefore we need the following definition. Definition 2.. Let Bz C[z ]. By Bz k C[z ] we nderstand a formal series sch that Bz k k = Bz, if sch a series Bz k exists. Lemma 2.. Let Az C[z ], ord A = and k N. There exists exactly k different determinations of Az k k, niqely determined by the choice a k their coefficient of z. Frthermore Az k k zc[z k ]. of

5 Characterization of Dhombres eqations 285 Proof. Let Cz C[z ] sch that 2.3 Az k = Cz k. Since ord Az k = k, we get ord C =, hence Cz = γ z + Cz with γ 0, C0 = 0. Write Az = a z + Ãz, Ã0 = 0, then and hence 2.3 a z k + Ãzk = γ k z k + Cz k, 2.3 { a = γ k, + Ãzk = + Cz k, C0 = 0. By the implicit fnction theorem we obtain that Cz exists and is niqely determined. Applying the binomial series + y k = + ν k ν y ν and sbstitting Ãzk for y we get + Cz = + Ãzk k = + k Ãz k ν C[z k ]. ν ν We have k different possibilities of γ = a k and frthermore Cz = γ z+ Cz zc[z k ]. To show the niqeness of C it is also possible to se other mehtods than the ones in the previos proof, therefore we have the following alternative proof. Proof of the niqeness of C. We start with Ãzk = + Cz k. Differentiating this leads to à z k kz k = k + Cz k C z which is eqivalent to à z k z k = + Cz k C z + Cz. By sbstitting 2.4 we obtain C à z k z k z = + Ãzk + Cz

6 286 L. Reich and J. Tomaschek or 2.5 à z k z k + Ãzk = C z + Cz, with C0 = 0. By Cachy s Theorem there exists a niqe Cz in a neighborhood of 0. Then à z k kz k = k + Cz k C z, implies Therefore we get d dz + Ãzk + Cz k = 0. + Ãzk + Cz k = δ C, where sbstitting z = 0 leads to δ = 0. This proves that C satisfies 2.4. In [4] it was shown that a series F C[z k ] if and only if F L η = F where L η z = ηz and η is a root of one of order k. We will se this in the proof of the following lemma. Lemma 2.2. Let F zc[z k ] with F z = ν=l β νk+z νk+. Then we have F z k C[z k ]. Proof. We write F z = β lk+ z lk+ + β νk z νk. ν= Let η be a root of one of order k, then sing η lk+ k = η lk η k = η k = we have F ηz k = βlk+ k η lk+ k z lk+ k k + β νk η νk z νk = = βlk+ k z lk+ k + = F z k ν= k β νk z νk = ν= and hence F z k C[z k ]. Since the following theorem depends on a special representation of the soltions of a generalized Dhombres fnctional eqation, it is necessary to consider the next lemma.

7 Characterization of Dhombres eqations 287 Lemma 2.3. Let F z C[z k ], F z = c k z k +..., c k 0 for some k N. Then there exists a series G C[z ] sch that Gc k z k = F z. Proof. We have F z = ν c νk z νk = = c k z k + ν 2 = Gc k z k. c νk c ν k c kz k ν = Together with the previos lemmata we can prove or main theorem. This theorem describes the solvability of a generalized Dhombres fnctional eqation. The proof is done in a similar way as it was done in [5]. Note that in this characterization we se the so-called generator g 0. Theorem 2.. Let w 0 C \ {0}. Then the transformed generalized Dhombres fnctional eqation 2. g = ϕg w 0 + g has a non constant soltion g if and only if there exists a fnction g 0, g 0 = +... and a k N sch that g ϕ = g 0 w 0 + k holds. Hence. has a non constant soltion f with f = w 0 {0, } if and only if there exists g 0 z = z +... and k N sch that 2.6 holds. Proof. Assme that there exists a g 0, g 0 = +... sch that ϕy = g 0 = g y 0 w 0+y holds. Then g k 0 is invertible and 2.6 becomes eqivalent to ϕ g 0 = g 0 w 0 + g 0 k. We apply the holomorphic transformation y = c k k, see [], and hence we get ϕ g 0 c k k c k k k = g 0 w 0 + g 0 c k k k = g 0 c k w 0 + g 0 c k k.

8 288 L. Reich and J. Tomaschek We define g := g 0 c k k and therefore we obtain 2. ϕg = g. w 0 + g In [6] we have shown that this eqation has non constant soltions g. Let s assme that the eqation g = ϕg has non constant w 0+g soltions g with ord g = k. Then it was shown before, that this eqation can be transformed to the linear fnctional eqation 2.2 w 0 + k U = Uψ. So we have to show that if 2.2 is solvable, then there exists a fnction g 0 sch that 2. holds. From [6] and by Lemma 2. the fnction ψ = w C[ k ]. We write U = U, U = +... and hence we obtain the eqivalent eqation or w 0 + w 0 k U = w 0 ψ U ψ + w 0 k ψ U = U ψ where ψ = +... C[ k ] Γ. Hence + w 0 k ψ = + which allows s to se the formal logarithm, we set A := Ln + w0 C[ k ] k ψ X := Ln U, ord X > 0. We obtain A + X = X ψ. Since A C[ k ] we decompose X in the following way, we write X = = X + X2 where X = ξ ν ν, X2 = ξ ν ν. Then we have ν ν 0 mod k ν ν 0 mod k Az = X ψ X + X 2 ψ X 2

9 Characterization of Dhombres eqations 289 where X ψ X C[ k ] and X2 ψ X2 C[ k ], if this expression is not zero. This leads to the system { A = X ψ X, = X 2 ψ X 2. The system 2.7 has a soltion X C[ k ] see [6]. Therefore U = = expx C[ k ] and after reversing or calclations also T C[ k ]. By Lemma 2.2 the fnction g = T k C[ k ] with g = = k + and hence by Lemma 2.3 there exists a fnction g k 0, g 0 y = y+... sch that gc k k = g 0 which proves this theorem. The sitation where f = redces to the sitation f0 = 0, see [6], and hence it is covered by [3]. 3. The case fz 0 =, z 0 0 As it is said in the introdction, in this sitation we do not obtain sch a general condition for solvability like in the previos one. We will se this section to show why this is even so. In fact the difference as it was said before lies in the missing of a generator g 0. In this case we also need some transformations, these are the following ones. Let f be a holomorphic soltion of the generalized Dhombres fnctional eqation. with fz 0 =, z 0 0 then we obtain the eqation 3. gz 0 gζ + ζ + ζgζ = ϕgζ where fz = w 0 + gz, gz 0 = 0 and gz = gz 0 + ζ =: gζ, gζ = c k ζ k +, k, c k 0 and ζ is defined as ζ = z z 0. Frthermore we se ϕw = ϕw 0 + ϕω with ω = w w 0 and ϕw 0 = 0 and ϕy = d m y m + with d m 0, for the details we refer the reader to [5] Section 4.. If we se T z k = gz and U = T eqation 3. becomes 3.2 z 0 ζ k + + ζ k Uζ = U ˆϕζ with ˆϕζ k = ϕζ k and ϕζ = d m ζ m + as we will see later. We have the following theorem. Theorem 3.. Let z 0 C, z 0 0, ϕy C[y ] with ϕ0 = 0. Let g C[ζ ], g 0 and ord g = k be a soltion of 3. gz 0 gζ + ζ + ζgζ = ϕgζ. Then ord g = k =.

10 290 L. Reich and J. Tomaschek Proof. We assme that k 2. Let gζ = c k ζ k +..., c k 0, then gz 0 gζ + +ζ+ζgζ = c k ζ k +... and hence ord gz 0 gζ+ζ+ζgζ = k. Eqation 3. implies ϕ 0 and ord ϕ, we have k = ord ϕ k which leads to ord ϕ = and therefore we write ϕy = d y + d 2 y , d 0. If we compare the coefficients of ζ k in 3. we obtain c k = d c k which gives s d =, becase c k 0. So we obtain ϕy = y + d 2 y We have ϕy y, becase if we assme that ϕy = y, 3. is given by gz 0 gζ + ζ + ζgζ = gζ. We write gζ = T ζ k with ord T =, this gives s T z 0 T ζ k + ζ + ζt ζ k k = T ζ k. After comparing the coefficients of ζ k we obtain We apply T on both sides, hence or T z 0 T ζ k + ζ + ζt ζ k = T ζ. ζ + z 0 T ζ k + ζt ζ = ζ z 0 + ζt ζ k = 0 which is a contradiction becase z 0 + ζ 0 and T ζ k 0. Hence ϕy = = y + d r y r +..., r 2, d r 0. Next we consider 3. with the transformation gζ = T ζ k, we have Sbstitting T ζ for ζ leads to T z 0 T ζ k + ζ + ζt ζ k k = ϕt ζ k. T z 0 ζ k + T ζ + ζ k T ζ k = ϕt ζ k. Then, by Lemma 2. there exists a niqely determined ψ C[z ] with ϕζ k = = ψζ k, ψy = y Writing U := T we obtain or This is the same as z 0 ζ k + + ζ k Uζ = Uψζ z 0 + ζ k + + ζ k Uζ = z 0 + Uψζ. + ζ k z 0 + Uζ = z 0 + Uψζ,

11 Characterization of Dhombres eqations 29 which is eqivalent to + ζ k + z0 Uζ = + z 0 Uψζ. We write V ζ := + z0 Uζ, then we have V ζ = +... and ζ k V ζ = V ψζ. Let V and V 2 be two soltions of 3.3, with V j ζ = +..., j =, 2. We define W ζ = Vζ V 2ζ. Eqation 3.3 implies 3.4 W ζ = W ψζ, W ζ = We have ψζ = ζ +..., ψζ ζ, and hence let ψ t t C be the analytic embedding of ψ. Then 3.4 implies W ζ = W ψ t ζ for all t C, see [5]. We differentiate this eqation with respect to t, hence we obtain 0 = W y ψ tζ ψ tζ. t We set t = 0 and so we have W y = 0. Therefore W = and hence V = V 2. We conclde that 3.3 has at most one soltion V with V 0 =. By Lemma 2. we know that ψζ ζc[ζ k ], ψζ = ζ Let ɛ = e 2πi k, we sbstitte ɛζ for ζ in 3.3, hence, with ψɛζ = ɛψζ we get + ζ k V ɛζ = V ɛψζ. Therefore V ɛζ is also a soltion of 3.3 with V ɛ 0 =. Hence V ζ = = V ɛζ, and so we have V ζ C[ζ k ]. We write V ζ = Ṽ ζk with Ṽ y C[y ]. We sbstitte this representation into 3.3, we get which is eqivalent to Therefore we have + ζ k Ṽ ζk = Ṽ ψζk + ζ k Ṽ ζk = Ṽ ϕζk. + yṽ y = Ṽ ϕy in C[y ] with Ṽ 0 =. We define Ln Ṽ =: X and hence we get Ln + y = X ϕy Xy with ord X. We have Ln + y = y + and therefore ord Ln + y = = y + =. Let ord X = l, then ord X ϕy Xy > l becase ϕy = y +... which is a contradiction. Hence k =.

12 292 L. Reich and J. Tomaschek We want to prove the following theorem in a different way than it was done in [5], becase then we can see the dependence of the soltions on the initial vale z 0. Theorem 3.2. Let ϕζ C[ζ ], ϕζ = d ζ +..., d 0. If d is no root of one, then 3.2 has a niqe soltion. Proof. We consider the fnctional eqation 3.2 for k =, compting both sides of 3.2, together with Uζ = ζ + 2 ζ +... leads s to z 0 + ζ ζ ζ n + n ζ n + = = d ζ + d d 2 ζ [P n,..., n + n d n ] ζ n +. Hence we obtain = d z 0 2 = P 2 d, d 2, z 0. n = P n d,..., d n, z 0. So the coefficients only depend on the coefficients of ϕ and on the vale z 0. If d is a root of nity, we have to distingish between the case where ϕ is linearizable and where it is not. We will not do this in this article. After the transformations which leads to eqation 3.3 we see, that we can immediately apply the formal logarithm to eqation 3.3 and we see, that there is no parameter like compare with the proof of Theorem 2. which cancels before only z 0 was removed. Exactly this is the difference between this case and the case before. We have no generator g 0 sch that we can represent all or soltions by g 0. Nevertheless in the sense of or solvability condiditons we can state the following two theorems. Note that k is always eqal to. Theorem 3.3. Let z 0 C \ {0} and let ϕζ C[ζ ], ϕζ = d ζ +..., d 0. Then the transformed generalized Dhombres fnctional eqation 3. gz 0 gζ + ζ + ζgζ = ϕgζ is solvable if and only if there exists an invertible g C[ζ ] sch that 3.5 ϕζ = gz 0 ζ + g ζ + ζg ζ. In more detail we can show that

13 Characterization of Dhombres eqations 293. if gζ = c ζ + sch that c z 0 + E and 3.5 holds, then 3. has a niqe soltion, 2. if gζ = c ζ + sch that c z 0 + E, ϕζ = c z 0 + ζ + is not linearizable and 3.5 holds, then 3. has a niqe soltion, 3. if gζ = c ζ + sch that c z 0 + E, ϕζ = c z 0 + ζ + is linearizable and 3.5 holds, then 3. has a soltion. We omit the proof of 2. and 3. becase of the length of it. Theorem 3.4. Let z 0 C \ {0} and let ϕζ C[ζ ], ϕζ = d m ζ m +, d m 0, m 2. Then the transformed generalized Dhombres fnctional eqation 3. gz 0 gζ + ζ + ζgζ = ϕgζ is solvable if and only if there exists an invertible g C[ζ ] sch that 3.6 ϕζ = gz 0 ζ + g ζ + ζg ζ. Proof. We want to give an idea of the proof. Let the transformed generalized Dhombres eqation be solvable. Then we transform it to 3.2 z 0 ζ + + ζuζ = U ˆϕζ, where we see that = z 0 and for 2 ν m we have ν = ν z 0. For ζ m we obtain m + m = z 0 d m. We proceed by indction. Reversing or calclations show that ord g =. On the other hand let ϕ be given by 3.6. Then we se the invertibility of g sch that we obtain the transformed generalized Dhombres eqation. This eqation is solvable. References [] Cartan, H., Elementare Theorien der Analytischen Fnktionen einer oder mehrerer Komplexen Veränderlichen, Bibliographisches Institt, Mannheim/Wien/Zürich 966. [2] Dhombres, J., Applications associatives o commtatives, C. R. Acad. Sci. Paris, Sr. A, ,

14 294 L. Reich and J. Tomaschek [3] Reich, L., J. Smítal and M. Štefánková, Local Analytic Soltions of the Generalized Dhombres Fnctional Eqation I, Sitzngsber. österr. Akad. Wiss. Wien, Math.-nat Kl. Abt. II, , [4] Reich, L., J. Smítal and M. Štefánková, Local Analytic Soltions of the Generalized Dhombres Fnctional Eqation II, J. Math. Anal. Appl., 355, No , [5] Tomaschek, J., Contribtions to the local theory of generalized Dhombres fnctional eqations in the complex domain, Grazer Math. Ber., , p. 72+iv. [6] Tomaschek, J. and L. Reich, Local soltions of the generalized Dhombres fnctional eqation in a neighborhood of infinity, sbmitted. L. Reich Institt für Mathematik nd Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Astria ldwig.reich@ni-graz.at J. Tomaschek Mathematics Research Unit Université d Lxemborg Ville de Lxemborg Grand Dchy of Lxemborg joerg.tomaschek@ni.l